3.2 An engineer's formulation

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3.2 An engineer's formulation

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After a section on what a physicist believes is a mathematician's view of the subject, it is time for an example. Using the format of a ``recipe'' that is applicable to a broad class of practical problems, we show here how the advection-diffusion equation (1.3.2#eq.2) is discretized using Galerkin linear finite elements (FEMs) and how it is implemented.

Derive a weak variational form.

``Multiply'' your equation by $\forall g\in \mathcal{C}^1(\Omega), \int_\Omega dx\; g^*(x)$ where the complex conjugate is necessary only if your equation(s) is (are) complex

$\displaystyle \forall g\in \mathcal{C}^1(\Omega),\hspace{1cm} \int_{x_L}^{x_R} ... ...+u\frac{\partial f}{\partial x} -D\frac{\partial^2 f}{\partial x^2} \right] = 0$

(1)

Integrate by parts,

so as to avoid having to use quadratic basis functions to discretize the second order diffusion operator

$\displaystyle \int_{x_L}^{x_R} dx \;\;\; \left[ g\frac{\partial f}{\partial t} ... ...ft.Dg\frac{\partial f}{\partial x}\right\vert _{x_L}^{x_R} = 0 \;\;\; \forall g$

(2)

Assuming a periodic domain, the surface term has here been canceled, imposing natural boundary conditions.

Discretize time

with a finite difference backward in time and using a partially implicit evaluation of the unknown $f=(1-\theta)f^t + \theta f^{t+\Delta t}$ where $\theta\in[1/2;1]$

$\displaystyle \int_{x_L}^{x_R} dx \left[ g\left(\frac{f^{t+\Delta t}-f^t}{\Delt... ...ial}{\partial x} \left[(1-\theta)f^t + \theta f^{t+\Delta t}\right] \right] = 0$

$\displaystyle \int_{x_L}^{x_R}\!\!dx \left[ \frac{g}{\Delta t} +\theta ug\frac{... ...!\theta) D \frac{\partial g}{\partial x}\frac{\partial}{\partial x} \right] f^t$

(3)

$\forall g$ . All the unknowns have been reassembled on the left. Re-scale by $\Delta t$ , and

Discretize space

using linear roof-tops and a Galerkin choice for the test functions

$\displaystyle f^t(x)=\sum_{j=1}^N f_j^t e_j(x), \hspace{2cm}\forall g\in\{e_i(x)\},\;\; i=1,N$

(4)

$\displaystyle \int_{x_L}^{x_R}\!\!$	$\displaystyle dx$	$\displaystyle \left[ e_i\sum_{j=1}^N f_j^{t+\Delta t} e_j +\Delta t\theta e_i u... ...tial x} \sum_{j=1}^N f_j^{t+\Delta t} \frac{\partial e_j}{\partial x} \right] =$
$\displaystyle \int_{x_L}^{x_R}\!\!$	$\displaystyle dx$	$\displaystyle \left[ e_i\sum_{j=1}^N f_j^{t } e_j +\Delta t(\theta-1) e_i u \su... ... e_i}{\partial x} \sum_{j=1}^N f_j^{t } \frac{\partial e_j}{\partial x} \right]$
		$\displaystyle \forall i=1,N$	(5)

Note how the condition $\forall g\in \mathcal{C}^1([x_L; x_R])$ is here used to create as many independent equations

as there are unknowns $\{f_j^{t+\Delta t}\}, j=1,N$ . All the essential boundary conditions are then imposed by allowing

and

to overlap in the periodic domain. Since only the basis and test functions

and perhaps the problem coefficients

remain space dependent, the discretized equations are all written in terms of inner products, involving overlap integrals of the form $(e_i, u e_j^\prime)=\int_{x_L}^{x_R}\!\!dx \;\; u(x) e_i(x)e_j^\prime(x)$ . Reassembling them all in a matrix,

Write a linear system

through which the unknown values from the next time step $f_j^{t+\Delta t}$ can implicitly be obtained from the current values

by solving a linear system

$\displaystyle \sum_{j=1}^N\mathbf{A_{ij}} f_j^{t+\Delta t} = \sum_{j=1}^N\mathbf{B_{ij}} f_j^t$

(6)

To relate the Galerkin linear FEM scheme (3.2#eq.5) with the code that has been implemented in the JBONE applet, it is necessary now to evaluate the overlap integrals. This is usually performed numerically using a quadrature. In the case of a homogeneous mesh $x_j=j\Delta x$ , a constant advection and diffusion coefficient , the coming section shows how explicit expressions can be obtained from the same rules to define directly the matrix elements.

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